Problem: Determine how many solutions exist for the system of equations. ${-4x-y = -5}$ ${y = -4x+5}$
Convert both equations to slope-intercept form: ${-4x-y = -5}$ $-4x{+4x} - y = -5{+4x}$ $-y = -5+4x$ $y = 5-4x$ ${y = -4x+5}$ ${y = -4x+5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+5}$ ${y = -4x+5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-4x-y = -5}$ is also a solution of ${y = -4x+5}$, there are infinitely many solutions.